A remark on double cosets. (Q2431540)

Let \(G\) be a virtually soluble group, and let \(A,B\leq G\) be finitely generated virtually Abelian subgroups of \(G\). In this paper it is proved (Theorem A) that if there are only finitely many double cosets \(AgB\) (\(g\in G\)), then \(G\) is virtually polycyclic. The special case \(A=B\) is connected to the question of orbits of groups of automorphisms: if \(G=N\rtimes A\), then \(A\backslash G/A\) is in one-to-one correspondence with the set of orbits of the action of \(A\) on \(N\). A key step in the proof of Theorem A is the following interesting result: ``Let \(A\) be a finitely generated virtually Abelian group, \(N\) a \(\mathbb ZA\)-module, and \(\delta\colon A\to N\) a derivation. If the resulting affine action of \(A\) on \(N\) has only finitely many orbits, then \(N\) is finitely generated as an Abelian group.''